what is the average of the points a, b and c with weights 1, 1 and 1 respectively?

1. First, assume the coordinates of the points:
   • Let's assume the coordinates of point (A=(1,4)), point (B=(2, - 7)), and point (C=(5,6)) (by observing the positions on the coordinate - plane).
   • For a weighted average of points in a two - dimensional plane, we calculate the weighted average of the (x) - coordinates and the weighted average of the (y) - coordinates separately.
   • The formula for the weighted average of a set of values (x_1,x_2,x_3) with weights (w_1,w_2,w_3) is (\bar{x}=\frac{w_1x_1 + w_2x_2+w_3x_3}{w_1 + w_2+w_3}).
2. Calculate the weighted average of the (x) - coordinates:
   • Here, (x_1 = 1), (x_2 = 2), (x_3 = 5), (w_1 = 1), (w_2 = 1), (w_3 = 4).
   • (\bar{x}=\frac{1\times1 + 1\times2+4\times5}{1 + 1+4}=\frac{1 + 2+20}{6}=\frac{23}{6}\approx3.83).
3. Calculate the weighted average of the (y) - coordinates:
   • Here, (y_1 = 4), (y_2=-7), (y_3 = 6), (w_1 = 1), (w_2 = 1), (w_3 = 4).
   • (\bar{y}=\frac{1\times4+1\times(-7)+4\times6}{1 + 1+4}=\frac{4-7 + 24}{6}=\frac{21}{6}=3.5).

Answer:

The weighted - average point is ((\frac{23}{6},3.5)) or approximately ((3.83,3.5))